Trigonometric inequalities

  • #1

Main Question or Discussion Point

Please help me to confirm, weather the following step is correct

[tex] |\gamma| \leq \cos (\beta) [/tex]
[tex] \arccos (|\gamma|) \leq \beta [/tex]

does taking the arccos() on both sides of equation changes the relational operator??
 

Answers and Replies

  • #2
tiny-tim
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welcome to pf!

hi junaidnawaz! welcome to pf! :smile:

(have a beta: β and a gamma: γ and a ≤ :wink:)

arccos is defined as being in [0,π)

so long as β is also in [0,π), your equations are the same (because cos is monotone in that region, and therefore so is arccos) :smile:
 
  • #3
Thx v much for your reply.

in my case, the range of parameters is as,
[tex] 0 \leq \beta \leq \pi /2 [/tex]
[tex]-1 \leq \gamma \leq +1 [/tex]

by taking arccos() on both-sides, would it change the operator (from [tex] \leq[/tex] to [tex] \geq[/tex] ) or would it remain same ??
 
  • #4
tiny-tim
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by taking arccos() on both-sides, would it change the operator (from [tex] \leq[/tex] to [tex] \geq[/tex] ) or would it remain same ??
oh, i missed that! :rolleyes:

yes, cos is decreasing, so the ≤ changes to ≥ :smile:

(but, eg, sin is increasing, so the ≤ would stay the same :wink:)
 
  • #5
Thank you.

if
[tex] |\gamma| \leq \cos( \beta ) [/tex]

then

[tex] x \leq \beta[/tex]

can i find "x", by keeping the RHS fixed to [tex] \beta[/tex]

is this possible to find x ?? by keeping RHS and relation operator the same ??
 
  • #6
tiny-tim
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sorry, i don't understand …

isn't x just |γ| ? :confused:
 
  • #7
[tex] |\gamma| \leq \cos ( \beta ) [/tex]

When I take arccos() on both sides, it becomes

[tex] \arccos( |\gamma| ) \geq \beta [/tex]

however, i want to keep [tex] \beta [/tex] on right side, and i want to keep the relational operator as [tex] \leq [/tex], i.e.,

[tex] x \leq \beta [/tex]

what would be x ??
 
  • #8
I wounder if its not a stupid question .... :P
 
  • #9
tiny-tim
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however, i want to keep [tex] \beta [/tex] on right side, and i want to keep the relational operator as [tex] \leq [/tex], i.e.,

[tex] x \leq \beta [/tex]
that's not possible (unless you replace β by some decreasing function of β, such as 1/β or -β)
 
  • #10
Thank you :)
 

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